R help - Oct 2011 - reasonable theory?

Before coding this in C, I wanted to test the idea out in R.

But I'm unsure if the theory is well-founded.

I have a (user-supplied) black-box function which takes R^n -> R^3
and a defined domain for each of the input reals.

I want to send some samples through the box to determine an
approximation of the convex hull of the function's range.
(I'll use the library from http://www.qhull.org to compute
the convex hull from the function's outputs.)

My plan is to use the permutation of the min and max values
for the n inputs along with k-1 samples w/in [min,max], but
I want the adjust the k samples a bit to avoid sampling bias.

To make it simpler, let's set the domain to [0,1].

Then, K = { 1/k, 2/k, ... (k-1)/k }

One reasonably easy possibility is to add to each Kn
a linear RV in, say, [-1/k?,1/k?].

Would a normal RV be better?  Some other bell-shaped RV?

Does adding a bit (but not too much) of randomness to the
input values have reason at all?

-JimC
-- 
James Cloos <cloos at jhcloos.com>         OpenPGP: 1024D/ED7DAEA6

If you know nothing about the black box except that its domain is 
bounded, then I would random sample uniformly from the domain.  If the 
function is monotonically increasing in all variables, then you only 
need to test the two extreme points.  If you know other things, you may 
be able to use the other logic.  Spencer


On 10/12/2011 1:42 PM, James Cloos wrote:
> Before coding this in C, I wanted to test the idea out in R.
>
> But I'm unsure if the theory is well-founded.
>
> I have a (user-supplied) black-box function which takes R^n ->  R^3
> and a defined domain for each of the input reals.
>
> I want to send some samples through the box to determine an
> approximation of the convex hull of the function's range.
> (I'll use the library from http://www.qhull.org to compute
> the convex hull from the function's outputs.)
>
> My plan is to use the permutation of the min and max values
> for the n inputs along with k-1 samples w/in [min,max], but
> I want the adjust the k samples a bit to avoid sampling bias.
>
> To make it simpler, let's set the domain to [0,1].
>
> Then, K = { 1/k, 2/k, ... (k-1)/k }
>
> One reasonably easy possibility is to add to each Kn
> a linear RV in, say, [-1/k?,1/k?].
>
> Would a normal RV be better?  Some other bell-shaped RV?
>
> Does adding a bit (but not too much) of randomness to the
> input values have reason at all?
>
> -JimC

-- 
Spencer Graves, PE, PhD
President and Chief Technology Officer
Structure Inspection and Monitoring, Inc.
751 Emerson Ct.
San Jos?, CA 95126
ph:  408-655-4567
web:  www.structuremonitoring.com